Re: Cerberus and Quine
Date: 23 Feb 2005 07:31:08 -0800
Paul Holbach wrote:
> > email@example.com wrote:
> > While Paul is obviously very right in saying
> > that truth does not have
> > to conform to common sense, I think especially
> > in caring about ordinary
> > language, you have to do ordinary philosophy of
> > language, and respect
> > common sense, because in a strong sense,
> > that is definitely what you
> > want to explain.
> I carefully chose my words when I said that common sense was *not
> always* the key to wisdom.
> I didnīt say "never"!
> > The true philosophy of mathematics
> > has to explain why and how axioms
> > are selected, regardless of this
> > confusing metaphysical talk.
> Apart from the methodology of mathematics and the philosophy of
> mathematics not being one and the same, the "true" philosophy of
> mathematics has to provide substantial and elaborate answers to the
> following crucial questions:
> "The job of the philosopher is to give an account of mathematics and
> its place in our intellectual lives.
> - What is the subject-matter of mathematics (ontology)?
> - What is the relationship between the subject-matter of mathematics
> and the subject-matter of science, which allows such extensive
> application and cross-fertilization?
> - How do we manage to do and know mathematics (epistemology)?
> - How can mathematics be taught?
> - How is mathematical language to be understood (semantics)?
> In short, the philosopher must say something about mathematics,
> something about the application of mathematics, something about
> mathematical language, and something about ourselves.
> A daunting task, even without the job of eliciting first principles."
> [Shapiro, Stewart (2000). /Thinking about mathematics: The
> of mathematics/. Oxford: Oxford University Press. (p. 15f)]
I agree with your explanations, Paul. We merely don't agree that
realism satisfies questions at all these levels. It seems, in
particular, on ontology.
It is a challenge that goes beyond the skills of even the finest
mathematicians and logicians to determine what is "philosophically"
true or significant about mathematics.
Let me point out my position briefly, so it becomes more apparent where
I stand. I would like a metaphysics which is compatible with the modern
view of the mind. This view excludes any kind of dualism and Platonism.
So, these are out of question for me. However, you can still be a
realist on some matters, you don't have to go all solipsist or
fictionalist (agnostic). What I say is that the Quine-Putnam argument
is not true as it stands, but there is indeed some objective truth to
mathematics which aliens would discover. The trouble is that, this
objective truth has many intriguing forms that might deceive a simple
metaphysics. For instance, a theory of infinity will have a necessarily
fictional component, on the other hand it will be an "objective
fiction" which can be analyzed with the scientific method, and it can
even prove use in scientific thought, *as long as it is also a proper
Anyway, the ontology I take is simple: substance monism, and a
mechanical world. I deal in nothing more than that, because I simply
don't believe in angels.
On epistemology, the language of philosophical logic has no first place
for me, but logic as method of philosophy, certainly has. I do not
think that one can access any truth except by experimentation, either
outwardly, or inwardly. I think even the notion of "truth" is an
abstraction we use to make sense of the world, for instance we like to
define it as correspondence to reality. But then, in our conversation,
I see that "reality" has degenerated into a mixture of observations,
story telling, and fantasizing, spanning a wide range of cognitive
activities. I don't find this talk scientific, especially when one is
talking about unscientific concepts like transfinite ordinals.
Therefore, I would like to first consider a positivistic account that
is free of some convictions of Quine: holism (non-explanation),
logicism (mixing knowledge with existence). I think logic is just that,
a way to represent and process your knowledge. But apparently predicate
logic is not the only way to do that, right? Anyway, there are
reductions among many formalisms, but this does not mean that one of
them is somehow special. It's just that whatever formalism "works", the
human keeps. Sometimes, these formalisms can be irrational. I don't
defend "irrationalism", but the situation is this: logic does not even
tell us why its premises are true, right? So, what does this tell you?
How do you know the premises are true? By many layers of cognitive
processing, that are simply not rational in the sense of logicism, or
even proper lambda calculus. If the mind is mechanical you can write a
logical translation of that mechanism in first-order logic, but I don't
think it's going to be any more rational than any "experiment". For the
most part, we ascertain truth to our observations due to statistical
computations that are actually going on in our mind, not scholastic
-- Eray Ozkural